Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 25, 1-6;http://www.math.u-szeged.hu/ejqtde/
An existence result of asymptotically stable solutions for an integral
equation of mixed type
Cezar AVRAMESCU1 and Cristian VLADIMIRESCU1
Abstract
In the present Note an existence result of asymptotically stable solutions for the integral equation
x(t) =q(t) +
Z t
0
K(t, s, x(s))ds+
Z ∞
0
G(t, s, x(s))ds
is presented.
2000 Mathematics Subject Classification: 47H10, 45D10. Key words and phrases: fixed points, integral equations.
1 Department of Mathematics, University of Craiova
13 A.I. Cuza Str., Craiova RO 200585, Romania
E-mail: zarce@central.ucv.ro, vladimirescucris@yahoo.com
1. Introduction
In this Note we will present an existence result of asymptotically stable solutions to the equation
x(t) =q(t) + Z t
0
K(t, s, x(s))ds+ Z ∞
0
G(t, s, x(s))ds, (1.1)
under hypotheses which will be given in Section 2. We call the integral equation (1.1) to be of mixed type, since within its form an operator of Volterra type and an operator of Uryson type appear. The notion of asymptotically stable solution to the functional equation
x=F(x) (1.2)
has been recently introduced in [6] and reconsidered in a more general framework in [7]. Let F : BC → BC be an operator, where BC := BCIR+,IRd
= {x : IR+ → IRd, x bounded and continuous}, IR+:= [0,∞), d≥1. Letx∈BC be a solution to Eq. (1.2).
Definition 1.1 The function x is said to be anasymptotically stable solution of (1.1) if for any ε >0
there exists T =T(ε)>0 such that for every t≥T and for every other solution y of (1.1), then
|x(t)−y(t)| ≤ε, (1.3)
where |·|denotes a norm in IRd.
Remark that in [6] the cased= 1 is considered, unlike [7] wherein the general case is treated.
In our papers [2]-[4] we studied the existence of asymptotically stable solutions for certain particular cases of Eq. (1.2), in which integral operators appear. Eq. (1.1) considered in the present Note is more general than those of [2]-[4].
2. Notations and preliminaries
Let |·| be an arbitrary norm in IRd, ∆ := {(t, s)∈IR
+×IR+, s≤t}. Admit that q : IR+ → IRd, K : ∆×IRd→IRd,G: IR+×IR+×IRd→IRd are continuous functions.
The proof of the existence of asymptotically stable solutions to Eq. (1.1) is divided in two steps. First, we show that (1.1) admits solutions and next we prove that there exist solutions fulfilling Definition 1.1.
Consider the functional space
Cc:= n
x: IR+ →IRd, x continuous o
,
equipped with the numerable families of seminorms
|x|n:= sup t∈[0,n]
{|x(t)|}, n≥1, (2.1)
or
|x|λn := sup t∈[0,n]
n
|x(t)|e−λnto
, (λn>0) n≥1. (2.2)
Each of these two families determine on Cc a structure of Fr´echet space (i.e. a linear, metrisable, and complete space), its topology being the one of the uniform convergence on compact subsets of IR+, for every sequenceλn. We also mention that a family A ⊂Cc is relatively compact if and only if for each n≥1, the restrictions to [0, n] of all functions fromA form an equicontinuous and uniformly bounded set.
3. Main result
In this section we will admit the following hypotheses:
(k) there exist continuous functionsα, β : IR+→IR+, such that
|K(t, s, x)−K(t, s, y)| ≤α(t)β(s)|x−y|,
for all (t, s)∈∆ and allx, y∈IRd;
(g) there exist continuous functions a, b: IR+→IR+, withR0∞b(t)dt <∞, such that
|G(t, s, x)| ≤a(t)b(s),
for all (t, s)∈∆ and allx∈IRd.
Lemma 3.1 Let z: IR+→IR+ be a continuous function, satisfying the condition
z(t)≤α(t) Z t
0
β(s)z(s)ds+γ(t), t∈IR+, (3.1)
where γ : IR+ →IR+ is continuous function. Then, there exists a continuous functionh : IR+ →IR+, such
that
z(t)≤h(t), ∀t∈IR+.
Proof. Let us denote
w(t) := Z t
0
β(s)z(s)ds. (3.2)
Then (3.1) becomes
z(t)≤α(t)w(t) +γ(t)
and, since (3.2),we obtain
w(0) = 0, w′
By (3.3), classical estimates lead us to conclude that
z(t)≤α(t)eR
t
0α(s)β(s)ds
Z t
0
β(s)γ(s)e−R
s
0 α(u)β(u)duds+γ(t) =:h(t), ∀t∈IR
+. (3.4)
✷
Definition 3.1 The operator H :Cc → Cc is called contraction if there is a sequence Ln ∈ [0,1), such
that
|Hx−Hy|λn ≤Ln|x−y|λn, ∀x, y∈Cc, ∀n≥1. (3.5)
Proposition 3.1 (Banach) Every contraction admits a unique fixed point.
The proof is classical and follows the proof of the known Banach’s Contraction Principle. We remark that the result still holds if (3.5) is fulfilled only on a closed set M, for whichH(M) ⊂M. Finally, notice that Proposition 3.1 is a particular case of a more general result due to Cain & Nashed (see [8]).
Proposition 3.2 ([9]) Let A, B:Cc →Cc be two operators fulfilling the following hypotheses:
(i) A is contraction;
(ii) B is compact operator;
(iii) the set
y=λA λy
+λBy, y∈Cc, λ∈(0,1) is bounded.
Then there exists x∈S, such thatx =Ax+Bx.
The result contained in Proposition 3.2 has been obtained in the case of a normed space by Burton & Kirk (see [5]) and it represents the generalization of a known theorem of Krasnoselskii. The result of Burton & Kirk has been extended in [1] in the case of a Fr´echet space.
Lemma 3.2 Admit that hypothesis (k) is fulfilled. Then the equation
x(t) =q(t) + Z t
0 K(t, s, x(s))ds, t∈IR+ (3.6)
admits a unique solution inCc.
Proof. We define the operatorH:Cc→Cc through
(Hx) (t) :=q(t) + Z t
0 K(t, s, x(s))ds, x∈Cc, t∈IR+.
Letn≥1 be fixed. Obviously, fort∈[0, n],
|(Hx) (t)−(Hy) (t)| ≤ α(t) Z t
0 β(s)|x(s)−y(s)|ds
≤ Lneλnt|x−y|λn,
whereLn= (1/λn) sup(t,s)∈∆n{α(t)β(s)},∆n={(t, s)∈[0, n]×[0, n], s≤t},and so
|Hx−Hy|λn ≤Ln|x−y|λn.
If we chooseλn>sup(t,s)∈∆n{α(t)β(s)}, it follows by Proposition 3.1 that Eq. (3.6) has a unique fixed
point. ✷
Theorem 3.1 Admit that hypotheses (k) and (g) are fulfilled. Then, Eq. (1.1) admits solutions in the set
(i) As in the proof of Lemma 3.2 it follows thatA is contraction.
(ii) We prove that B is compact operator.
First, since hypothesis (g), the convergence of the integral R∞
0 G(t, s, y(s) +ξ(s))ds is uniform with
and the continuity ofB is proved.
LetS ⊂Cc be bounded andn≥1 be fixed. Then,∃pn>0,∀x∈ S,|x|n≤pn. Clearly, for allt∈[0, n] and y∈ S, we have
|(By) (t)| ≤an Z ∞
So,nBy|[0,n], y∈ S
References
[1] C. Avramescu, Some remarks on a fixed point theorem of Krasnoselskii,Electronic Journal of Qualitative Theory of Differential Equations, 5, 1-15 (2003).
[2] C. Avramescu, C. Vladimirescu, On the existence of asymptotically stable solutions of certain integral equations, in preparation.
[3] C. Avramescu, C. Vladimirescu, Asymptotic stability results for certain integral equations, in prepara-tion.
[4] C. Avramescu, C. Vladimirescu, Remark on Krasnoselskii’s fixed point Theorem,in preparation.
[5] T.A. Burton and C. Kirk, A fixed point theorem of Krasnoselskii type,Mathematische Nachrichten,189, 23-31 (1998).
[6] J. Bana´s and B. Rzepka, An application of a measure of noncompactness in the study of asymptotic stability, Applied Mathematics Letters,16, 1-6 (2003).
[7] T.A. Burton and Bo Zhang, Fixed points and stability of an integral equation: nonuniqueness,Applied Mathematics Letters,17, 839-846 (2004).
[8] G.L. Cain, Jr. and M.Z. Nashed, Fixed points and stability for a sum of two operators in locally convex spaces, Pacific Journal of Mathematics,39(3), 581-592 (1971).
[9] M.A. Krasnoselskii, Some problems of nonlinear analysis,American Mathematical Society Translations, Ser. 2, 10(2), 345-409 (1958).
